random vertices up-Z/down-Z only..?
Note: I assume you want to randomize the vertex positions of a mesh object.

Displace modifier

One solution, that will work on any mesh is to use the 'Displace' modifier.


Add a 'Displace' modifier to the mesh object you want to add noise to. Set 'Direction' to 'Z' and add a texture (use the big new button).





Open the texture tab, and change the type of the texture to 'Noise' (if you want uniform noise, use clouds with small scale instead).





Play around with the 'Strength' property of the modifier, until you get the needed result (Circled in the first image).


Randomize tool

Alternatively, you can use the 'Randomize' Mesh editing tool if all your normals are on ±Z (e.g. a grid). To do this, make sure that you set the 'normal' option to 1.0 (boxed on the image below).